= Solution
{c}
Take outward radial <volume flux per unit width> as positive. The <streamfunction in polar coordinates> immediately gives
$$
q(r)=\int_0^\alpha v_r(r,\theta)r\,d\theta
=\psi(r,\alpha)-\psi(r,0)
=\boxed{-\frac12r^2\dot\alpha.}
$$
For spanwise width $W$, the <volumetric flow rate> is $Q=Wq$. The wedge inside radius $r$ has cross-sectional <area> $\alpha r^2/2$, whose rate of increase is $r^2\dot\alpha/2$. Opening therefore draws in exactly the <volume> needed to fill the expanding wedge; closing expels it. This is also a direct check of <conservation of mass> for the moving <control volume>. It follows from the wall <boundary conditions> and remains true even though the printed interior <streamfunction> is erroneous.
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